Theorem ralrab2 3372

Description: Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ralab2.1	|- ( x = y -> ( ps <-> ch ) )

Assertion

Ref	Expression
ralrab2	|- ( A. x e. { y e. A | ph } ps <-> A. y e. A ( ph -> ch ) )

Distinct variable groups:   x, y   x, A   ch, x   ph, x   ps, y

Allowed substitution hints:   ph( y)   ps( x)   ch( y)   A( y)

Proof of Theorem ralrab2

Step	Hyp	Ref	Expression
1		df-rab 2921	. . 3 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2	1	raleqi 3142	. 2 |- ( A. x e. { y e. A | ph } ps <-> A. x e. { y | ( y e. A /\ ph ) } ps )
3		ralab2.1	. . 3 |- ( x = y -> ( ps <-> ch ) )
4	3	ralab2 3371	. 2 |- ( A. x e. { y | ( y e. A /\ ph ) } ps <-> A. y ( ( y e. A /\ ph ) -> ch ) )
5		impexp 462	. . . 4 |- ( ( ( y e. A /\ ph ) -> ch ) <-> ( y e. A -> ( ph -> ch ) ) )
6	5	albii 1747	. . 3 |- ( A. y ( ( y e. A /\ ph ) -> ch ) <-> A. y ( y e. A -> ( ph -> ch ) ) )
7		df-ral 2917	. . 3 |- ( A. y e. A ( ph -> ch ) <-> A. y ( y e. A -> ( ph -> ch ) ) )
8	6, 7	bitr4i 267	. 2 |- ( A. y ( ( y e. A /\ ph ) -> ch ) <-> A. y e. A ( ph -> ch ) )
9	2, 4, 8	3bitri 286	1 |- ( A. x e. { y e. A | ph } ps <-> A. y e. A ( ph -> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   {cab 2608   A.wral 2912   {crab 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921

This theorem is referenced by:  efgsf  18142  ghmcnp  21918  nmogelb  22520  pntlem3  25298  sstotbnd2  33573